Calculate Deltoid Perimeter: Given Area b² Square Centimeters

Let's find the perimeter of deltoid ABCD with given area $b^2$.

The formula for the area of a kite (or deltoid) is $\frac{1}{2} \times d_1 \times d_2 = b^2$. Here $d_1$ and $d_2$ are the lengths of the diagonals of the kite.
Since $\frac{1}{2} \times d_1 \times d_2 = b^2$, we rearrange to find $d_1 \times d_2 = 2b^2$.

In a typical kite, each diagonal is the perpendicular bisector of the other; therefore, each side of the kite can be derived using diagonals through their perpendicular intersections.

If we take the segments created by the intersection of the diagonals, from the Pythagorean theorem, each pair of equal kite sides involves terms of the form $\sqrt{a^2 + b^2}$, derived from these segments.

For example, if this kite is specifically structured such that the diagonals are split into segments where $d_1 = b\sqrt{2}$ and $d_2 = b\sqrt{5}$, then:

• Each pair of sides derived from these diagonal arrangements will also utilize $\sqrt{\left(\frac{b\sqrt{2}}{2}\right)^2 + \left(\frac{b\sqrt{5}}{2}\right)^2 }$.
• Therefore, the calculation leads us to $b\sqrt{2} + b\sqrt{5}$ for each side.

Combining this for a total of four sides of the kite (two of each equal one):

Perimeter = $2(b\sqrt{2} + b\sqrt{5}) = 2b(\sqrt{2} + \sqrt{5})$.

Thus, the perimeter of deltoid ABCD is $2b(\sqrt{2} + \sqrt{5})$.